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OPEN entanglement of a harmonic oscillator with an electromagnetic feld Received: 23 November 2017 Dmitry N. Makarov Accepted: 2 May 2018 At present, there are many methods for obtaining of with an Published: xx xx xxxx electromagnetic feld. Most methods have a low probability of quantum entanglement and not an exact theoretical apparatus based on an approximate solution of the Schrodinger equation. There is a need for new methods for obtaining quantum-entangled particles and mathematically accurate studies of such methods. In this paper, a quantum harmonic oscillator (for example, an in a magnetic feld) interacting with a quantized electromagnetic feld is considered. Based on the exact solution of the Schrodinger equation for this system, it is shown that for certain parameters there can be a large quantum entanglement between the electron and the electromagnetic feld. Quantum entanglement is analyzed on the of a mathematically exact expression for the Schmidt modes and the Von Neumann .

It is known that quantum entanglement arises when the of a system cannot be represented as a product of the wave functions of each particle. Tere are many such systems from which to choose. Te basic method of quantum entanglement of particles is spontaneous parametric down-conversion (SPDC)1–3. In this process, when a nonlinear birefringent crystal is irradiated by a laser pump feld the pump decay with a low probability into two quantum-entangled photons of lower frequencies. Despite the low probability of the decay of the pump , SPDC has been widely studied and has many practical applications in quantum com- puter science. Tis is due to the fact that this method is the simplest for obtaining quantum-entangled photons. While other more sophisticated methods of generating entangling particles have been proposed4–8 none of them result in a greater probability of entanglement of the particles under consideration than SPDC9,10. Tere is also a difculty with the theoretical study of the various methods, which is that basically, quantum entanglement is a complex process for the mathematical research. For the systems under consideration, it is impossible to solve exactly the nonstationary Schrodinger equation and calculate the entanglement parameter (for example, the in the case of a two-component system). Solutions are mainly sought using higher orders of perturbation theory and on the basis of these solutions conclusions are drawn about quantum entanglement. It is known that perturbation theory is applicable if there is a weak interaction of particles in the system, and therefore use of this theory is based on the assumption that quantum entanglement is not a large quantity. In order to study a system with large quantum entanglement, the exact solution of the Schrodinger equation must be used - a task that is very difcult to achieve mathematically. Nonetheless, when studying quantum entanglement, such a solu- tion would be of fundamental interest. In quantum physics, it is known that exact solutions of the Schrodinger equation do present any great difculty. In studying quantum entanglement, one can include a two-level model of Jaynes-Cummings11 in order to achieve sufciently accurate solutions. In this model are studied the physical characteristics of a two-level atom in an electromagnetic feld. It should also be noted that the Jaynes-Cummings model describes the system of a two-level atom interacting with a quantized electromagnetic feld, which greatly limits its application to many physical systems (the results presented in this work go beyond this limitation). Also some model problems and particular cases for the stationary Schrodinger equation12,13. of quantum-entangled systems is being studied actively at the present time14–17. For example, on the basis of the exact solution of the nonstationary Schrodinger equation for an atom in a strong two-mode electromagnetic feld, it has been shown17,18 that there is a strong quantum entanglement between photons. However, as this solution is based upon the assumption that the external electromagnetic feld is many times stronger than the Coulomb feld of attraction of an electron in an atom, it does not promote an understanding of the infuence of the Coulomb

Northern (Arctic) Federal University named after M.V. Lomonosov, Arkhangelsk, 163002, Russia. Correspondence and requests for materials should be addressed to D.N.M. (email: [email protected])

Scientific REPOrts | (2018)8:8204 | DOI:10.1038/s41598-018-26650-8 1 www.nature.com/scientificreports/

Figure 1. Interaction of an electromagnetic feld with an electron in a magnetic feld.

feld on quantum entanglement. Terefore, the problem of fnding a system whose quantum entanglement is large, and also in which it would be possible to mathematically study quantum entanglement on the basis of an exact solution of the Schrodinger equation, is topical. In addition to this relevance, the problem of practical appli- cation is important, i.e. the proposed method should be simple enough for practical implementation. It should be added that quantum entanglement is widely used in quantum science. Tat is why at present a great interest is attracted to this problem. In particular, using entangled states, it is possible to explain quantum communication protocols19, such as quantum cryptography20, quantum coding21, quantum key distri- bution22, quantum secure direct communication23, learning24,25, entanglement swapping26,27, quantum algorithms28 and teleportation29. Tere are quite a lot of practical problems in quantum informatics, one of which is the weak generation of quantum-entangled particles. Te method of generation of quanta-entangled particles proposed in this work gives a large amount of quantum entanglement. In this paper we study the generation method of quantum entanglement of a harmonic oscillator with an external electromagnetic feld. A harmonic oscillator can be an electron in a uniform magnetic feld with induc- tion B, see Fig. 1. It is known30 that in such a feld the electron behaves like a harmonic oscillator with frequency Be ωc = , where c is the , me and −e are, the mass and charge of an electron, respectively. Te wave mce function of such a system can be found on the basis of the exact solution of the Schrodinger equation of the spe- cifc system being studied. Te quantum entanglement of the system can be studied using the analytical form of the Schmidt mode31,32 and the von Neumann entropy33,34. It is shown that for certain parameters of the system under consideration, a large quantum entanglement is possible. Exact solution of the Schrodinger equation Let us consider, at a time t > 0, a quantized electromagnetic feld acting on a system consisting of a quantum harmonic oscillator. Te interaction of a harmonic oscillator of mass m occurs via an electric charge e. Te Schrodinger equation for this system will be

 2 22 22 ∂Ψ  1  ∂ e  mxωδc ((++yz)) i=  −i + Aˆ  ++Hˆ Ψ.    f  ∂tm2 ∂r c 2  (1) In the expression: (1) δ = 0 corresponds to one type of interaction, and δ = 1 to another type; i is the imaginary unit; ω is the frequency of the oscillator; Hˆ = ∑ ω aa+ + 1 is the Hamiltonian of the electromagnetic c f ku, ˆˆku,,ku 2 + () feld, where aˆku, and aˆku, are the creation and annihilation operators, respectively, for photons with the wave vec- tor k and the u; and the vector potential of the electromagnetic feld Aˆ has the form

2 2πc  ⁎ + Auˆ =−∑ ((expi()ωωtakr )(ˆˆku,,+−ukexpi()ta−.r ))ku ku, ωVf (2)

Te summation for Hˆ f and expression (2) include all possible values of the wave vector k and the polarization u. Next we consider a single-mode feld, in which ∑ku, has one summand with the wave vector k with polarization u. Further, for convenience we apply a system of units similar to atomic ħ = 1, m = 1, e = 1. It should be added that any suitable physical system can be an oscillator. In such a system, the charge e and the mass m depends on the

Scientific REPOrts | (2018)8:8204 | DOI:10.1038/s41598-018-26650-8 2 www.nature.com/scientificreports/

problem under consideration. Terefore, everywhere below, under the oscillator interacting with the electromag- netic feld, we mean any suitable physical system, the Schrodinger equation of which is described by the expres- sion (1). Now consider expression (1) in the dipole approximation in which the vector potential, expressed in 2 terms of feld variables, is equal to Auˆ = aq, where a = 4πc , ω is the frequency of the mode under considera- ωV tion, V is the quantization volume, and q is the feld variable of the mode. Since we are considering a single-mode electromagnetic feld, it is convenient to orient the polarization vector along the x axis i.e. u = i, where i is the unit vector directed along the x axis (the same assumption applies throughout the paper). For example, expression (1) for δ = 0 corresponds to the case of an electron in a magnetic feld interacting with an external quantized electromagnetic feld if the polarization of the electromagnetic feld u is in the plane perpendicular to the magnetic induction vector B. Tis is easily obtained by using the Landau gauge30 for the magnetic feld AL = Bxj, where j is the unit vector directed along the y axis. As a result, the dynamics of the system, and electronic transitions are not determined by the stationary Schrodinger equation with the Hamiltonian

2 1  ∂  ω  ∂2  ω 22x Hiˆ = − + βqq +  2 −  + c ,    2  22 ∂x   ∂q  2 (3)

where β = 4π . It should be clarifed that expression (3) corresponds to the Hamiltonian without the variables ωV y, z, since the wave function corresponding to the solution of equation (1) Ψ(x, y, z, q, t) = Ψ(y, z, t)Ψ(x, q, t), and this means that Ψ(y, z, t) is not a perturbed wave function and therefore does not afect the electronic transitions of the system under the action of an external electromagnetic field. Extending the argument, we replace x → ωc x, and the Hamiltonian (3) becomes ω  ∂2  ω  ∂2  β2 ∂ Hqˆ =  2 −  + c x2 −  +−qi2 βωq .  2   2  c 22 ∂q   ∂x  2 ∂x (4) We consider the nonstationary Schrodinger equation with the Hamiltonian (4). We represent the diferential equation in question in the form ∂Ψ′ Hiˆ ′Ψ′=  , ∂t (5) −1 −1 where Hˆˆ′=SHˆˆS and Ψ′ =ΨSˆ , with Sˆ being a unitary and Sˆ being the inverse operator of Sˆ (i.e. −1 SˆˆS = 1). Finding the solution of the Schrodinger equation (5), we obtain the required wave function using the −1 inverse transformation Ψ=Sˆ Ψ′. We seek a solution for Ψ′ in the form, where the operator    ∂ ∂  Seˆ = xpiγαexpi()qx .  ∂xq∂  (6) It should be noted that the choice of the operator Sˆ in expression (6) is determined as a result of a careful anal- ysis of the stationary Schrodinger equation with the Hamiltonian (4) for the possibility of its diagonalization. In expression (6), the constant values γ, α are unknown: our task now is to fnd values of γ, α so that the Hamiltonian Hˆ ′ in (5) becomes diagonal. Knowing the form of the operator Sˆ in expression (6) and using known properties of the operators ˆˆ (eAAHeˆˆ− =+HA[,ˆˆHA]1++/2[,ˆˆ[,AHˆ ]] ) it is not difcult to obtain the diagonal Hamiltonian Hˆ ′; all that is required is to fnd the appropriate parameters γ, α. As a result, we get

ω 1 ω 1 ωω22−+βω2 α =+c εε 2 1 , γ =± , ε = c . ( ) 2 ω 2 ωc ε + 1 2βωωc (7) When β → 0, the system must go to the initial state, for this it is necessary to use the upper sign in expression (7) (and elsewhere below) for ε > 0, and the lower sign when ε < 0. Te case with ε = 0 will be considered below. As a result, the expression for Hˆ ′ takes the form

ω  ∂2  ω  1 ∂2  Hqˆ ′= Λ−2 σ  + c  x2 − κ ,  2   2  22 ∂q  σ ∂x  (8)

2 2ωc 2βα ωc β 1 Λ=1 +−α +=, σ ω , ω ω ω 1 + α2 ωc 22βεγ 2 βγ κσ=+ −+(1 αγ). ω ωc (9) It is not difcult to solve the Schrodinger equation with the Hamiltonian (8), since all the variables are separated. −′iEnm, t Te solution for the nonstationary equation will be sought in the form Ψ′(,xq,)tA= ∑nm, nm,,exΨnm(,q),

Scientific REPOrts | (2018)8:8204 | DOI:10.1038/s41598-018-26650-8 3 www.nature.com/scientificreports/

′′ where Hˆ ′Ψnm,,(,xq)(=ΨExnm nm, ,)q , n,0m =…,1,2, are quantum numbers, An,m are coefcients that depend on the initial conditions, and En,m are the eigenvalues of the energy. First, we write the eigenvalue of the energy of the Hamiltonian (8)      1   1  Ennm, = ωωc +  Gm+  +  S ,  2   2  (10) where

βω βω β2 GS= 1 ,1=− c εε2 ++.1 2 ω3/2 ( ) ω 21ωσc ε + (11) ′′′ Next, we write out the wave function of the Hamiltonian (8) in the form Ψ=nm, (,xq)(ΨΨnmxq)(), where

′−R x 22′−s q Ψ=nn()xCeH22nm()xR,(Ψ=qC)(meHm qs), (12)

1 Λ and where Hn, Hm are Hermite polynomials, R = , s = , and the normalization coefcients σκ σ

1 1/411 /4 Cn ==RC, m s . 2!nn ππ2!mm (13)

−1 − ˆ iEnm, t We then fnd Ψ=(,xq,)tSΨ′(,xq,)t . As a result, Ψ=(,xq,)tA∑nm, nm,,exΨnm(,q), where the wave − −1 1 ′ ˆ ′ function Ψ=nm, (,xq)(Sxˆ Ψnm, ,)q . Te action of the operator S on the function Ψnm, (,xq) is not an obvious ′ problem. For this, for example, the wave function Ψn()x must be represented in the form of a Fourier integral ∞ Ψ=′ ()xa1 ()pexp ip 1 xdp, where for the inverse Fourier transform we obtain nn2π ∫−∞ σκ () −1 n 2 an(p) = Cn(−i) exp(−p /2) and only then and only then can the operator Sˆ act. As a result, we get

n 2 ∞ p s 2 CCnm()−i −−ix()pR αγq −+()qp R Ψ=nm, (,xq) eH2 n()pe e 2 2π ∫−∞

×+Hsm((qpγ Rd)) p. (14) Function (14) can be calculated analytically using the table integral35, but the result is complicated enough without doing further analysis. It will be shown later that for calculating probabilities and quantum entanglement, the analytic form of the function (14) is not required, so this calculation is not given here. ∞ Consider the Fourier integral for the wave function Ψ(x, q, t) in the form Ψ=(,pq,)tx1 Ψ(,qt,) 2π ∫−∞ exp()ipxdx. Afer a simple calculation we fnd that

− Ψ= iEnm, tΨ (,pq,)tA∑nm, nm,,epnm(,q), where

1/4   1 2   n s  ()pq−α  1  Ψ=nm, (,pq) ccnmi   eH2R n ()pq− α  R   R  −+s ((qp1)αγ −γ )2 ×+eH2 m((sq(1 αγ))− γp ), (15) n −−11m and where cn ==(2nc!)ππ,(m 2!m ) . We show that the Fourier transform of the wave function can also be used, like the wave function itself, to calculate the probabilities. We expand the wave function in the −−iE t iE t eigenfunctions of the unperturbed system in the form Ψ=(,xq,)ta∑ ()txΦΦ()eqmm1 ()e 2 , mm12, mm12, m1 m2 where EE, are the energy of states, with quantum numbers m , m , respectively, for a noninteracting oscilla- mm12 1 2 tor in the state |m and a free electromagnetic feld in the |m and at() is the probability ampli- 1〉 2〉 mm12, tude for fnding the system in state ΦΦ()xq,() at time t. By the defnition of probability w =|at()|2 mm12 mm12,,mm12 is the probability of detecting a system in the state m1, m2 at time t. Carrying out the Fourier transform over this −− m1 iEmm12tiEt decomposition (as shown above), we obtain Ψ(,pq,)t = i ∑mm, atmm, ()eeΦΦmm()pq(). Using the 12 12miEtiE t 12 orthogonality condition, we obtain at() = ()−ie1 mm12e ΦΦ()pq()|Ψ(,pq,)t = mm12, mm12 m −ΔiE t −−iE t iE t ()−iA1,∑ e nm ΦΦ()pe mm1 ()qe 2 |Ψ (,pq) , where Δ=EEE−−E . nm, nm, m1 m2 nm, nm,,nm mm12 In order to calculate at() must be known An,m; these are sought from the initial conditions of the problem ∞ mm12, Ψ=(,pq,0)(1 Ψ xq,,0)expi()px dx. Assume that at the initial instant of time t = 0 the system was in the 2π ∫−∞ state Ψ=(,xq,0)(ΦΦxq)(). Te initial state of the system is a noninteracting oscillator in the state |s and a ss12 1〉 free electromagnetic feld in the Fock state |s . As a result, Ψ=(,pq,0)(ips1ΦΦ)(q) and using the orthogonal- 2〉 ss12 ity properties of Ψ|nm,,(,pq)(Ψ=nm′′pq,) δδnn,,′′mm (δnn, ′ is the Kronecker symbol), we obtain

Ai=Ψs1 (,pq)(|Φ pq)(Φ.) nm,,nm ss12 (16)

Scientific REPOrts | (2018)8:8204 | DOI:10.1038/s41598-018-26650-8 4 www.nature.com/scientificreports/

If in the expression (16) the coefcient An,m is renamed so that it refects information about the state from ss12, which it is calculated AAnm,,→ nm, we then obtain −Δ at()=.AAss12, ⁎mm12, e iEnm, t mm12, ∑ nm, nm, nm, (17) Next, we make a simplifcation, based on the fact that the parameter β entering expression (17) is a small quantity. Indeed, for a realistic microcavity or focal volume36, this value is of the order of 10−5–10−3, although it is usually even less. It can be seen from expressions (7), (9) and (15) that the quantum entanglement of the system will be signifcant if the parameter ε is a fnite value. Since β  1, and ε is a fnite quantity, we get ω ≈ ωc ≈ ω, and Δω = ω − ωc is less than or of order β. We write out expression (15) retaining the main terms under the condition β  1 and ω ≈ ωc ≈ ω  σ  n  2 Ψ=nm, (,pq) icnmcexp−−()pqα   2   σ   2 ×−Hpnm((σαqe)) xp−+()qpασHq((+ αp)),  2  (18)

where σ in this case will be σ = (1 + α2)−1, and α =+εε 2 1. Te parameter α is the main parameter of the problem under consideration and α ∈ (−1, 1), and σ ∈ (1/2, 1). It should be noted that the wave function (18) can be obtained by rotating the orthogonal coordinate system [the coordinates (p, q) → (p′, q′), where p′=pqcossθθ− in , and qq′= cossθθ+ p in ] by an angle θ corresponding to the value tgθ = α. In order to calculate (16) with the wave function (18), the results of previous research17 must be used where an integral of this kind was calculated. As a result, we obtain

sn12−+smsm2+  2  in(1− )!α m! −++− 2 + α ss12, = ((1)ss12,)ns2 −  Anm, ss+ Pm  2 , 2 12  α  (1 + α )!2 ss12! (19) (,bc) where Pxa () is the Jacobi polynomial. In expression (19), as was shown in the previous study in calculating the 17 integral , the condition s1 + s2 = m + n is fulfilled. If the condition s1 + s2 = m + n is satisfied, then m1 + m2 = m + n is executed, which means s1 + s2 = m1 + m2. Tis is a very important condition; as will be shown below, its use permits the calculation of the Schmidt modes. Tus, the for detecting a system in the state |m1〉 |m2〉 during the evolution of the system from the state |s1〉 |s2〉 is obtained in the analytical form and is determined by the expressions (10), (19) and

ss12+ ss, ⁎mm, −ΔiE t at()=.AAe12 12 ns, 12+−sn mm12, ∑ ns, 12+−snns, 12+−sn n=0 (20) In the expression (20), up accurate to an inessential phase, ΔE for β  1 can be replaced by ns, 12+−sn Δ→Enδ , where δβ=+ωα()ε . In addition, the wave function Ψ(x, q, t) is accurate to an inessential ns, 12+−sn phase, and taking into account the fact that s1 + s2 = m1 + m2, it can be represented in the form

ss12+ Ψ=(,xq,)ta ()txΦΦ() ()q . ∑ ms11, +−sm21 ms11+−sm21 m1=0 (21)

Schmidt’s Modes and Quantum Entanglement According to Schmidt’s theorem31,32, the wave function of the system can be expanded in the form

Ψ=∑k λkkux()vqk(), where uk(x) is the wave function of the pure state 1 of the system, vk(p) is the wave func- tion of the pure state 2 of the system, and where λk is the Schmidt mode and is an eigenvalue of the reduced den- ρ ′= λ ⁎ ′ ρ ′= λ ⁎ ′ λ sity matrix, i.e. 1(,xx)(∑k kkux)(uxk ) or 2(,qq)(∑k kkvq)(vqk ). Having found the Schmidt mode k, the measure of the quantum entanglement of the system can be calculated. To do this, various measures of entan- 31,32 21− 33,34 glement can be used, for example, the Schmidt parameter K = ()∑k λk or Von Neumann entropy SN =−∑k λλkkln(). The main difficulty in calculating quantum entanglement is the search for λk for a quantum-dynamic system. Using expression (21), the defnition of the reduced can be obtained

ss12+ ρ (,xx′=)(||at)(2 ΦΦxx)(⁎ ′), 1 ∑ ms11, +−sm21 mm11 m1=0 ss12+ ρ (,qq′=)(||at)(2 ΦΦqq)(⁎ ′), 2 ∑ ms11, +−sm21 ss12+−ms11+−sm21 m1=0 (22)

2 + from which it follows that λ =|at()| , and S =−∑ss12λλln(). kk,ss12+−k N k=0 kk We now present the results of quantum entanglement for the Von Neumann entropy SN. As an example, in Fig. 2 we consider the dependences of SN = SN(δt) on the dimensionless parameter for α = (1, 3/4, 1/2, 1/10, 1/100) and four variants of photon numbers s2 and oscillator s1. Calculation of entropy, for negative values of α, is

Scientific REPOrts | (2018)8:8204 | DOI:10.1038/s41598-018-26650-8 5 www.nature.com/scientificreports/

Figure 2. Te results of calculating the Von Neumann entropy SN = SN(δt) for α = (1, 3/4, 1/2, 1/10, 1/100) and (a) s1 = 0, s2 = 10; (b) s1 = 5, s2 = 10; (c) s1 = 10, s2 = 10; (d) s1 = 20, s2 = 10.

not necessary, since λk is an even function λk(α) = λk(−α). From this it follows that λk, and hence the probability |at()|2, of detecting a system in the state |m |m is a continuous function; nonetheless, α for ε = 0 is inter- mm12, 1〉 2〉 2 rupted. Indeed, if λk(α) = λk(−α), then α can be replaced by α → |α|, where |αε|= +−1 ||ε , which is a continuous function. Conclusion It can be seen from formula (20) and Fig. 2 that the entropy SN = SN(δt) is a 2π periodic function, i.e. SN(δt) = SN(δt + 2π). It is also interesting to note that when δ ~ β, the value is small; therefore, in order for the system to have a large quantum entanglement, it takes t ~ 1/β. The system will quickly go into a quantum-entangled state in the case when ωc is distinguishable from ω, but when β  1 from expression (7) it is clear that α  1 and γ  1, so the quantum entanglement will be small. As a result, we can say that to achieve a large quantum entanglement of the system, ωc ≈ ω is necessary, with Δω = ωc − ω less or about β, but the time for large quantum entanglement must be large t ~ 1/β. In the opposite case (ωc, as distinct from ω), the system quickly becomes a quantum-entangled state, but the quantum entanglement is very small. Quantum entanglement will be a large value in the case of s12+ s  1, for example, when the number of photons is large. It is known that the Von Neumann entropy has the maximum value max{SN} = ln N, where N is the number of nonzero eigenvalues; in our case, therefore, max{SN} = ln(s1 + s2) (which is well illustrated by the Fig. 2). Te advantage of the problem under investigation is that the quantum entanglement is not limited to the framework of the model and can take arbitrarily large values for a certain choice of s1 and s2. Te presented method can be an intensive source of quantum-entangled with photons. To charac- terize the intensity (or probability) of quantum-entanglement, the Schmidt parameter or the Von Neumann entropy is most ofen used. For example, if we compare the quantum entanglement on the basis of the Schmidt parameter, we get that the presented method can give more orders of magnitude more quantum entanglement than at SPDC. Indeed, the Schmidt parameter for SPDC in practice can be no more than ten37, in theory can 38 6 6 reach several hundred , but in our case this parameter is unlimited. For example, for s1 ~ 10 or s2 ~ 10 , and also α ~ 1, the Schmidt parameter is easy to calculate and at the maximum it can reach K ~ 106. It should also be added that the study of the quantum entanglement of the system under consideration is possible only on the basis of an exact solution of the Schrodinger equation (1). Other approaches, for example, using the Jaynes-Cummings model to solve the problem under consideration would not be possible, since our task is multilevel. An experimental realization of this method of quantum entanglement of an electron with an elec- tromagnetic feld is possible. To do this, it is sufcient to place an electron in a magnetic feld and irradiate such a system with a non-classical electromagnetic feld. Te methods for obtaining nonclassical states of the electromagnetic feld are well known. In order for quantum entanglement to be essential, it is necessary to select the parameters of the electromagnetic feld so that α ~ 1 (see (7)) for ω ≈ ωc, and also to choose large values of quantum numbers s1 and s2.

Scientific REPOrts | (2018)8:8204 | DOI:10.1038/s41598-018-26650-8 6 www.nature.com/scientificreports/

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Additional Information Competing Interests: Te author declares no competing interests. Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional afliations.

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